In the oil industry, it is common to be faced with a set of data from which one wishes to infer some sort of information of interest. It is also fairly common that such inverse problems are non-unique, that is, different solutions explain the data equally well. While it is straightforward to obtain a single solution that the user considers “most likely”, it is often desirable to know the “best” and “worst” case solutions that fit the data in addition to the “most likely” one, to adequately understand the risks of a given course of action. An example of this sort of problem in the oil industry is the prediction of the sand and porosity distribution in a reservoir where one would like to know the largest and smallest hydrocarbon volumes (i.e., the best and worst case scenarios) possible in addition to the “most likely”. An accurate understanding of the potential risks involved in draining a potential reservoir should reduce total costs (correctly sized platforms, optimal draining strategy, etc.).
A common method for determining alternative scenarios is to do forward simulations of many different models in which the variables deemed to affect the final result are chosen at random from some pre-defined distribution. The forward models are then compared to the observed data to see which of the various forward models match. A distribution of the parameters fitting the data is then extracted from the set of models that are deemed to fit the data well. From this distribution, a best and worst case can, in principle, be determined. This method is time-consuming because it requires a large number of forward models. In addition, it suffers from user bias in that the only models tried are the ones that the user has thought of or deemed relevant.
Another method is to take the most likely model and simply scale it up and down by some amount and call that the best and worst case. This method produces results that generally do not match the observed data in a forward modeling sense and are not necessarily the “best” and “worst” case answers.
What is needed is a method in which the best and worst case scenarios are obtained as mathematical solutions to the inverse problem.